Definition 2.1 Let \(m\in {\mathbb {Z}}_{\ge 2}\) . For \(\theta _k\in  (\) 1km\(),and \) k= (k(1), ..., k(N)) N\(,\) k=0, ..., m\(, such that \) j=1N k(j)=0\(,we define the \) m+2\( point \) q\(-conformal block by\begin{gather}\mathcal {F}\left({{\phantom{000} \theta _m\phantom{h000}\theta _{m-1}\phantom{h0}\phantom{h0}\cdots \phantom{h0}\phantom{h0}\theta _1\phantom{h0}}\atop {\sigma _{m}\phantom{h0}\sigma _{m-1}\phantom{h0}\sigma _{m-2}\phantom{h0}\cdots \phantom{h_0}\sigma _{1}\phantom{h0}\sigma _{0}}};\,x_m,\dots ,x_{1}\right)\nonumber \\ \qquad =\prod _{p=1}^{m}{\mathcal {N}}\Bigl ({\hspace{-1.0pt}\theta _p\atop {\sigma _{p} \phantom{h_2} \sigma _{p-1}}}\Bigr )q^{N\theta _p\Delta _{\sigma _{p}}}\prod _{p=1}^{m}x_p^{\Delta _{\sigma _{p}}-\Delta _{N\theta _p h_1}-\Delta _{\sigma _{p-1}}}\nonumber \\ \qquad \phantom{=}{}\times \sum _{{\lambda }_1,\dots , {\lambda }_{m-1}\in {\mathbb {Y}}^N}\prod _{p=1}^{m-1} \bigg (\frac{q^{N\theta _p} x_p}{x_{p+1}} \bigg )^{\left|{\lambda }_p\right|}\frac{\prod _{p=1}^m\prod _{k,k^{\prime }=1}^{N}N_{\lambda _p^{(k)}, \lambda _{p-1}^{(k^{\prime })}}\big (q^{\sigma _p^{(k)}-\theta _p-\sigma _{p-1}^{(k^{\prime })}}\big )}{\prod _{p=1}^{m-1}\prod _{k,k^{\prime }=1}^{N} N_{\lambda _p^{(k)}, \lambda _{p}^{(k^{\prime })}}\big (q^{\sigma _p^{(k)}-\sigma _{p}^{(k^{\prime })}}\big )}, \end{gather}where \) p=(p(1),...,p(N))\(,and \) 0=m=(, ..., )\(.\) 
Our \(q\) -conformal blocks can be obtained as expectation values of intertwining operators of the Ding–Iohara–Miki algebra [1]}. We omit a derivation and refer to [2]}.If we replace \(\mathcal {V}_{\theta ,w}\)  on p. 6 in [2]} by the tensor product of the \(N\)  Fock modules of the Ding–Iohara–Miki algebra, then we will get (), provided that the normalization is suitably chosen. The series part of a \(q\) -conformal block is the 5d Nekrasov instanton partition function.
